Results: Plots and Tables

This report contains different plots and tables that may be relevant for analysing the results. Observe:

• The hypercube \(Y_N\) is placed in, is approx. \([0, 10000m]^p\), since for each subproblem \(Y_N^s \subset [0, 10000]^p\). That is, more subproblems create a larger hypercube for the master problem and the volume (number of integer points) grows exponentially in \(p\).
• Given \(|Y_N|\), we can find the hypercube as \[[l_1, u_1] \times \cdots \times [l_p, u_p],\] where \(l_i = \min_{y \in Y_N}(y_i)\) and \(u_i = \max_{y \in Y_N}(y_i)\).
• The number of integer points in the hypercube is \[\prod_{i = 1}^p (u_i - l_i + 1)\]

Statistics for the problems solved using alg1

Given a problem consisting of \(m\) subproblems with \(Y_N^s\) given for each subproblem \(s\), we use a filtering algorithm to find \(Y_N\) (alg1).

Note that the width of objective \(i\), \(w_i = [l_i, u_i]\) should be approx. \(10000m\):

## # A tibble: 3 × 6
##       m mean_width1 mean_width2 mean_width3 mean_width4 mean_width5
##   <dbl>       <dbl>       <dbl>       <dbl>       <dbl>       <dbl>
## 1     2      19347.      19300.      19337.      19184.      18974.
## 2     3      27966.      28031.      27494.      27717.      26398.
## 3     4      38029.      38259.      37875.        NaN         NaN

Size of \(Y_N\)

What is \(|Y_N|\) given the different methods of generating the set of nondominated points for the subproblems?

## # A tibble: 4 × 3
##   method mean_card     n
##   <chr>      <dbl> <int>
## 1 l         40915.   110
## 2 m         45295.   110
## 3 u         39512.   110
## 4 ul        38208.   110

Does \(p\) have an effect?

## # A tibble: 16 × 4
## # Groups:   method [4]
##    method     p mean_card     n
##    <chr>  <dbl>     <dbl> <int>
##  1 l          2     2781.    30
##  2 m          2     2260.    30
##  3 u          2      708.    30
##  4 ul         2      717.    30
##  5 l          3    10765.    30
##  6 m          3     9864.    30
##  7 u          3     3093.    30
##  8 ul         3     3998.    30
##  9 l          4    63447.    25
## 10 m          4    84820.    25
## 11 u          4    66024.    25
## 12 ul         4    62641.    25
## 13 l          5   100325.    25
## 14 m          5    99928.    25
## 15 u          5   103271.    25
## 16 ul         5    99817.    25

Does \(m\) have an effect?

## # A tibble: 12 × 4
## # Groups:   method [4]
##    method     m mean_card     n
##    <chr>  <dbl>     <dbl> <int>
##  1 l          2    45336.    80
##  2 m          2    53458.    80
##  3 u          2    44689.    80
##  4 ul         2    43026.    80
##  5 l          3    38087.    20
##  6 m          3    31557.    20
##  7 u          3    37214.    20
##  8 ul         3    35707.    20
##  9 l          4    11207.    10
## 10 m          4     7464.    10
## 11 u          4     2700.    10
## 12 ul         4     4666.    10

Relative size of \(Y_N\)

Nondominated points classification

We classify the nondominated points into, extreme, supported non-extreme and unsupported.